241102 Elliptic Curves (V+Ü) (SoSe 2025)

Elliptic curves are fundamental objects in geometry and number theory. Although they are classical objects, elliptic curves remain central to modern research. They played a key role in the proof of Fermat's Last Theorem and are at the heart of one of the unsolved "Millennium Problems", the Birch and Swinnerton-Dyer conjecture, which concerns the study of their rational points.
The goal of this course is to provide a comprehensive introduction to both the geometric and arithmetic theory of elliptic curves. After an overview of algebraic curves, we will focus on the geometry of elliptic curves: Weierstraß equations, the group law, maps between elliptic curves, and their endomorphism groups.
Next, we examine the set of rational points on elliptic curves over finite fields, local fields, and global fields. A key result is the Mordell-Weil theorem, which states that the group of rational points over a global field is finitely generated. However, determining this group is not trivial and is closely linked to the above mentioned Millennium Problem.
Finally, we will explore further advanced topics, which may include L-series or moduli spaces of elliptic curves. The specific topics will be tailored to the interests of the participants.